Dirichlet's unit theorem

Last updated December 30, 2024

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet.^[1] It determines the rank of the group of units in the ring $O K$ of algebraic integers of a number field $K$ . The regulator is a positive real number that determines how "dense" the units are.

where $r 1$ is the number of real embeddings and $r 2$ the number of conjugate pairs of complex embeddings of $K$ . This characterisation of $r 1$ and $r 2$ is based on the idea that there will be as many ways to embed $K$ in the complex number field as the degree $n=[K:\mathbb {Q} ]$ ; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that

n = r 1 + 2 r 2

.

Note that if $K$ is Galois over $\mathbb {Q}$ then either $r 1 = 0$ or $r 2 = 0$ .

Other ways of determining $r 1$ and $r 2$ are

use the primitive element theorem to write $K=\mathbb {Q} (\alpha )$ , and then $r 1$ is the number of conjugates of $α$ that are real, $2 r 2$ the number that are complex; in other words, if $f$ is the minimal polynomial of $α$ over $\mathbb {Q}$ , then $r 1$ is the number of real roots and $2r 2$ is the number of non-real complex roots of $f$ (which come in complex conjugate pairs);
write the tensor product of fields $K\otimes _{\mathbb {Q} }\mathbb {R}$ as a product of fields, there being $r 1$ copies of $\mathbb {R}$ and $r 2$ copies of $\mathbb {C}$ .

As an example, if $K$ is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of Pell's equation.

The rank is positive for all number fields besides $\mathbb {Q}$ and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a determinant called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when $n$ is large.

The torsion in the group of units is the set of all roots of unity of $K$ , which form a finite cyclic group. For a number field with at least one real embedding the torsion must therefore be only ${1,-1}$ . There are number fields, for example most imaginary quadratic fields, having no real embeddings which also have ${1,-1}$ for the torsion of its unit group.

Totally real fields are special with respect to units. If $L / K$ is a finite extension of number fields with degree greater than 1 and the units groups for the integers of $L$ and $K$ have the same rank then $K$ is totally real and $L$ is a totally complex quadratic extension. The converse holds too. (An example is $K$ equal to the rationals and $L$ equal to an imaginary quadratic field; both have unit rank 0.)

The theorem not only applies to the maximal order $O K$ but to any order $O \subset O K$ .^[2]

There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of $S$ -units , determining the rank of the unit group in localizations of rings of integers. Also, the Galois module structure of $\mathbb {Q} \oplus O_{K,S}\otimes _{\mathbb {Z} }\mathbb {Q}$ has been determined.^[3]

The regulator

Suppose that K is a number field and $u_{1},\dots ,u_{r}$ are a set of generators for the unit group of K modulo roots of unity. There will be $r + 1$ Archimedean places of K, either real or complex. For $u\in K$ , write $u^{(1)},\dots ,u^{(r+1)}$ for the different embeddings into $\mathbb {R}$ or $\mathbb {C}$ and set $N j$ to 1 or 2 if the corresponding embedding is real or complex respectively. Then the $r \times (r + 1)$ matrix $\left(N_{j}\log \left|u_{i}^{(j)}\right|\right)_{i=1,\dots ,r,\;j=1,\dots ,r+1}$ has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries in a row). This implies that the absolute value $R$ of the determinant of the submatrix formed by deleting one column is independent of the column. The number $R$ is called the regulator of the algebraic number field (it does not depend on the choice of generators $u i$ ). It measures the "density" of the units: if the regulator is small, this means that there are "lots" of units.

The regulator has the following geometric interpretation. The map taking a unit $u$ to the vector with entries ${\textstyle N_{j}\log \left|u^{(j)}\right|}$ has an image in the $r$ -dimensional subspace of $\mathbb {R} ^{r+1}$ consisting of all vectors whose entries have sum 0, and by Dirichlet's unit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice is $R{\sqrt {r+1}}$ .

The regulator of an algebraic number field of degree greater than 2 is usually quite cumbersome to calculate, though there are now computer algebra packages that can do it in many cases. It is usually much easier to calculate the product $hR$ of the class number $h$ and the regulator using the class number formula, and the main difficulty in calculating the class number of an algebraic number field is usually the calculation of the regulator.

Examples

A fundamental domain in logarithmic space of the group of units of the cyclic cubic field K obtained by adjoining to
Q
{\displaystyle \mathbb {Q} }
a root of f(x) = x + x - 2x - 1. If a denotes a root of f(x), then a set of fundamental units is {e1, e2}, where e1 = a + a - 1 and e2 = 2 - a. The area of the fundamental domain is approximately 0.910114, so the regulator of K is approximately 0.525455. Discriminant49CubicFieldFundamentalDomainOfUnits.png — A fundamental domain in logarithmic space of the group of units of the cyclic cubic field $K$ obtained by adjoining to $\mathbb {Q}$ a root of $f (x) = x + x - 2 x - 1$ . If $α$ denotes a root of $f (x)$ , then a set of fundamental units is ${ε 1, ε 2}$ , where $ε 1 = α + α - 1$ and $ε 2 = 2 - α$ . The area of the fundamental domain is approximately 0.910114, so the regulator of $K$ is approximately 0.525455.

The regulator of an imaginary quadratic field, or of the rational integers, is 1 (as the determinant of a $0 \times 0$ matrix is 1).
The regulator of a real quadratic field is the logarithm of its fundamental unit: for example, that of $\mathbb {Q} ({\sqrt {5}})$ is ${\textstyle \log {\frac {{\sqrt {5}}+1}{2}}}$ . This can be seen as follows. A fundamental unit is ${\textstyle ({\sqrt {5}}+1)/2}$ , and its images under the two embeddings into $\mathbb {R}$ are ${\textstyle ({\sqrt {5}}+1)/2}$ and ${\textstyle (-{\sqrt {5}}+1)/2}$ . So the $r \times (r + 1)$ matrix is $\left[1\times \log \left|{\frac {{\sqrt {5}}+1}{2}}\right|,\quad 1\times \log \left|{\frac {-{\sqrt {5}}+1}{2}}\right|\ \right].$
The regulator of the cyclic cubic field $\mathbb {Q} (\alpha )$ , where $α$ is a root of $x 3 + x 2 - 2 x - 1$ , is approximately 0.5255. A basis of the group of units modulo roots of unity is ${ε 1, ε 2}$ where $ε 1 = α 2 + α - 1$ and $ε 2 = 2 - α 2$ .^[4]

Higher regulators

A 'higher' regulator refers to a construction for a function on an algebraic $K$ -group with index $n > 1$ that plays the same role as the classical regulator does for the group of units, which is a group $K 1$ . A theory of such regulators has been in development, with work of Armand Borel and others. Such higher regulators play a role, for example, in the Beilinson conjectures, and are expected to occur in evaluations of certain $L$ -functions at integer values of the argument.^[5] See also Beilinson regulator.

Stark regulator

The formulation of Stark's conjectures led Harold Stark to define what is now called the Stark regulator, similar to the classical regulator as a determinant of logarithms of units, attached to any Artin representation.^[6]^[7]

$p$ -adic regulator

Let $K$ be a number field and for each prime $P$ of $K$ above some fixed rational prime $p$ , let $U P$ denote the local units at $P$ and let $U 1, P$ denote the subgroup of principal units in $U P$ . Set $U_{1}=\prod _{P|p}U_{1,P}.$

Then let $E 1$ denote the set of global units $ε$ that map to $U 1$ via the diagonal embedding of the global units in $E$ .

Since $E 1$ is a finite-index subgroup of the global units, it is an abelian group of rank $r 1 + r 2 - 1$ . The $p$ -adic regulator is the determinant of the matrix formed by the $p$ -adic logarithms of the generators of this group. Leopoldt's conjecture states that this determinant is non-zero.^[8]^[9]

Notes

↑ Elstrodt 2007 , §8.D
↑ Stevenhagen, P. (2012). Number Rings (PDF). p. 57.
↑ Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.
↑ Cohen 1993 , Table B.4
↑ Bloch, Spencer J. (2000). Higher regulators, algebraic $K$ -theory, and zeta functions of elliptic curves. CRM Monograph Series. Vol. 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.
↑ Prasad, Dipendra; Yogonanda, C. S. (2007-02-23). A Report on Artin's holomorphy conjecture (PDF) (Report).
↑ Dasgupta, Samit (1999). Stark's Conjectures (PDF) (Thesis). Archived from the original (PDF) on 2008-05-10.
↑ Neukirch et al. (2008) p. 626–627
↑ Iwasawa, Kenkichi (1972). Lectures on $p$ -adic $L$ -functions. Annals of Mathematics Studies. Vol. 74. Princeton, NJ: Princeton University Press and University of Tokyo Press. pp. 36–42. ISBN 0-691-08112-3. Zbl 0236.12001.

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References

Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics. Vol. 138. Berlin, New York: Springer-Verlag. ISBN 978-3-540-55640-4. MR 1228206. Zbl 0786.11071.
Elstrodt, Jürgen (2007). "The Life and Work of Gustav Lejeune Dirichlet (1805–1859)" (PDF). Clay Mathematics Proceedings. Archived from the original (PDF) on 2021-05-22. Retrieved 2010-06-13.
Lang, Serge (1994). Algebraic number theory. Graduate Texts in Mathematics. Vol. 110 (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94225-4. Zbl 0811.11001.
Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001

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[1] Elstrodt 2007 , §8.D

[2] Stevenhagen, P. (2012). Number Rings (PDF). p. 57.

[FOOTNOTENeukirchSchmidtWingberg2000proposition_VIII.8.6.11-3] Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.

[4] Cohen 1993 , Table B.4

[Bloch-5] Bloch, Spencer J. (2000). Higher regulators, algebraic $K$ -theory, and zeta functions of elliptic curves. CRM Monograph Series. Vol. 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.

[6] Prasad, Dipendra; Yogonanda, C. S. (2007-02-23). A Report on Artin's holomorphy conjecture (PDF) (Report).

[7] Dasgupta, Samit (1999). Stark's Conjectures (PDF) (Thesis). Archived from the original (PDF) on 2008-05-10.

[NSW6267-8] Neukirch et al. (2008) p. 626–627

[9] Iwasawa, Kenkichi (1972). Lectures on $p$ -adic $L$ -functions. Annals of Mathematics Studies. Vol. 74. Princeton, NJ: Princeton University Press and University of Tokyo Press. pp. 36–42. ISBN 0-691-08112-3. Zbl 0236.12001.

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